Central Force Fields
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Central force fields are widely used in physics and they have simple yet important properties.
In general, central force is described using
.. math::
\vec F(\vec r) = f(r)\hat r.
The Lagrangian for an object of mass :math:`m` in a central force field is
.. math::
L &= \frac{1}{2} m \mathbf{ \dot r} ^2 - V(r) \\
& = \frac{1}{2} m ( \mathbf{\dot r}^2 + r^2 \theta^2 ) - V(r) .
The interesting thing for such a system is that there is always a conserved quantity since the Lagrangian has no explicit :math:`\theta` dependence. It is obvious that
.. math::
\frac{\partial L}{\partial \theta} = 0.
Now we have
.. math::
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}} \right) = 0,
which leads to the conservation of angular momentum as the first equation of motion,
.. math::
\dot l \equiv \dot p_\theta = \frac{d}{dt} \left( m r^2 \dot\theta \right) = 0
The second equation of motion is given by
.. math::
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{r}} \right) - \frac{\partial L}{\partial r} = 0,
which simplifies to
.. math::
\frac{d}{dt} (m \dot r) - m r {\dot \theta}^2 + \frac{\partial V(r)}{\partial r} = 0.
Applying the conserved quantity, we find an effective potential
.. math::
V_{eff} (r) = V(r) + \frac{1}{2} \frac{ l^2 }{ m r^2 }.